Problem 29 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3,-1) and (2,4)

Step-by-Step Solution

Answer

The point-slope form of the line is \( y - 4 = x - 2 \) and the slope-intercept form is \( y = x + 2 \)

Step 1: Calculate the Slope

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). By substituting the provided points (-3,-1) and (2,4) into this formula, the slope of the line can be determined: \( m = \frac{4 - (-1)}{2 - (-3)} = \frac{5}{5} = 1 \).

Step 2: Construct the Point-Slope Form

The point-slope form of a line is given by the formula \( y - y_1 = m (x - x_1) \). Substituting \( m = 1 \) from step 1, and the given point (2,4) into this formula, the point-slope form of the line becomes: \( y - 4 = 1 (x - 2) \) or \( y - 4 = x - 2 \). This simplifies to \( y = x + 2 \)

Step 3: Construct the Slope-Intercept Form

Now with the point-slope form of the line (**y = x + 2**), it is easy to convert it into the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this case, the slope \( m \) is already known to be 1, and the intercept \( b \) is 2. Thus, the slope-intercept form of the line is \( y = 1x + 2 \), which can be simplified to \( y = x + 2 \)

Key Concepts

Point-Slope Form

The point-slope form of a line is a very handy way to write linear equations when you know the slope and one point on the line, which is especially useful if you don't have the y-intercept right away. This equation looks like this: \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \((x_1, y_1)\) stands for the coordinates of the known point you've learned.

• Use this form when you are given a point and a slope but not the y-intercept.
• It helps to provide the structure to easily rearrange into other forms, like the slope-intercept form.

Once you plug in the values you gather, like from our problem with point (-3, -1) or (2, 4) and a slope of 1, it becomes straightforward to rearrange for any other format or solve for coordinates along the line.

Slope-Intercept Form

The slope-intercept form of a line might be the most recognized equation. This form is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept, or the point where the line crosses the y-axis.

• This is often the easiest form to use if you need to graph a line or quickly understand the slope and intercept.
• The y-intercept gives you the 'starting point' of the line on a graph, making it very intuitive.

This format provides a clear picture of how the line behaves across a graph. From our exercise with a slope of 1 and y-intercept 2, the equation \( y = x + 2 \) quickly helps visualize how the line rises and where it touches the y-axis.

Calculating the Slope

The slope, or gradient, of a line is a measure that tells you how steep the line is. It can be calculated using two points on the line with the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on your line.

• Slope indicates both the direction and steepness of a line. A positive slope means the line rises as it moves left to right; a negative slope means it falls.
• A slope of 0 signifies a horizontal line, while an undefined slope is characteristic of a vertical line.

In our problem, with points (-3, -1) and (2, 4), applying the slope formula gives a slope of 1. This signifies a 45-degree angle rise, signifying a direct one-to-one increase in both x and y dimensions.